High probability generalization bounds for uniformly stable algorithms with nearly optimal rate

Algorithmic stability is a classical approach to understanding and analysis of the generalization error of learning algorithms. A notable weakness of most stability-based generalization bounds is that they hold only in expectation. Generalization with high probability has been established in a landmark paper of Bousquet and Elisseeff (2002) albeit at the expense of an additional $\sqrt{n}$ factor in the bound. Specifically, their bound on the estimation error of any $γ$-uniformly stable learning algorithm on $n$ samples and range in $[0,1]$ is $O(γ\sqrt{n \log(1/δ)} + \sqrt{\log(1/δ)/n})$ with probability $\geq 1-δ$. The $\sqrt{n}$ overhead makes the bound vacuous in the common settings where $γ\geq 1/\sqrt{n}$. A stronger bound was recently proved by the authors (Feldman and Vondrak, 2018) that reduces the overhead to at most $O(n^{1/4})$. Still, both of these results give optimal generalization bounds only when $γ= O(1/n)$. We prove a nearly tight bound of $O(γ\log(n)\log(n/δ) + \sqrt{\log(1/δ)/n})$ on the estimation error of any $γ$-uniformly stable algorithm. It implies that for algorithms that are uniformly stable with $γ= O(1/\sqrt{n})$, estimation error is essentially the same as the sampling error. Our result leads to the first high-probability generalization bounds for multi-pass stochastic gradient descent and regularized ERM for stochastic convex problems with nearly optimal rate --- resolving open problems in prior work. Our proof technique is new and we introduce several analysis tools that might find additional applications.